{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R346",
  "content_hash": null,
  "slug": "gtc-claim-alpha-growth",
  "type": "claim",
  "title": "Trees with no alpha-labeling: the count doubles, the share peaks and falls",
  "summary": "Counted exhaustively: n=7: 1, n=8: 2, n=9: 5, n=10: 11, n=11: 22, n=12: 44, n=13: 82, n=14: 147, n=15: 254. The count grows by about 1.8 per vertex, while the share of all trees peaks at 10.6% at n=9 and falls to 3.3% at n=15.",
  "relevance": "For Graceful tree conjecture, record gtc-claim-alpha-growth (“Trees with no alpha-labeling: the count doubles, the share peaks and falls”) records a bound, answer, status fact, or structural consequence. The record states: Counted exhaustively: n=7: 1, n=8: 2, n=9: 5, n=10: 11, n=11: 22, n=12: 44, n=13: 82, n=14: 147, n=15: 254.",
  "relevance_source": "recorded",
  "body": "Counts of trees on \\(n\\) vertices with no alpha-labeling, from exhaustive search: n=7: 1, n=8: 2, n=9: 5, n=10: 11, n=11: 22, n=12: 44, n=13: 82, n=14: 147, n=15: 254. As a share of all trees of that size: n=7: 9.1%, n=8: 8.7%, n=9: 10.6%, n=10: 10.4%, n=11: 9.4%, n=12: 8.0%, n=13: 6.3%, n=14: 4.7%, n=15: 3.3%.\n\nThe two series move in opposite directions, and the distinction decides what an alpha-based argument can claim. The absolute count grows by about 1.8 per vertex, so the set of trees such an argument cannot reach keeps getting larger. The share peaks at 10.6% around n=9 and then falls, reaching 3.3% by n=15, so those trees are becoming rarer among all trees of their size.\n\nWhether the share continues down is open, and the answer is what would turn an alpha-based method into an asymptotic result. The measured range is short.",
  "status": "reproduced",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "trees on 7 to 15 vertices",
    "bounds": {
      "n": {
        "max": 15,
        "min": 7
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1403.1564",
      "locator": "research/graceful-trees/checks.py --max-n 15"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1403.1564",
    "locator": "research/graceful-trees/checks.py --max-n 15"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R342",
      "title": "Prove the conjecture by induction through alpha-labelings",
      "object_type": "attempt",
      "relation": "constrains",
      "direction": "outgoing"
    },
    {
      "slug": "gtc-problem-alpha-density",
      "title": "Do alpha-free trees become asymptotically rare?",
      "object_type": "problem",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "R338",
      "title": "Trees with no alpha-labeling through n=15",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "graceful-tree-conjecture",
      "title": "graceful tree conjecture",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}